Dividend Modeling. Forschungsseminar Stochastische Analysis und Stochastik der Finanzmärkte Humboldt University, Technical University Berlin

Transcription

1 J.P. Morgan Inroducon no Quanave esearch vdend Modelng Forschungssemnar ochassche Analyss und ochas der Fnanzmäre Humbold Unversy, Techncal Unversy Berln ecember 6, 2 Hans Buehler, JP Morgan Eques Q Q U A N T I T A T I V E E E A C H

2 Q U A N T I T A T I V E E E A C H vdends Am of hs al Par I General srucure of soc prce models wh dvdends for dervaves prcng Par II Wor owards a smple model for prcng opons drecly lned o dvdends 2

3 Par I General srucure of soc prce models wh dvdends for dervaves prcng

4 Q U A N T I T A T I V E E E A C H vdends Bascs ocs may pay dvdends; no oblgaon o do so. Mos companes ry o manan a seady ncome sream bu ulmaely, he dvdend paymen depends on he performance of he company ze of dvdend may change Tmng of dvdend may change pecal dvdends Tmelne. Announcemen of he dvdend amoun 2. All holders of he shares a he ex-dvdend dae are enled o receve he dvdend 3. vdend s pad a paymen dae and seled a selemen dae 4

5 Q U A N T I T A T I V E E E A C H vdends Bascs In a classc frconless mare, he paymen of a dvdend has o be neural o he holder of he share: he prce of he share jus before he dvdend has o be equal o he prce jus afer he dvdend plus he dscouned dvdend paymen. oherwse ener he respecve arbrage poson. If he ex-dvdend dae s, he selemen dae s s and he (random) dvdend s, hs means (*) F(, s) We denoe by - he lef hand lm of n. vdend should be - measurable and posve. 5

6 Q U A N T I T A T I V E E E A C H vdends Bascs Can we observe hs drop n soc prce n pracce? No: vdends go ex overngh whle here s no radng. The volaly n he openng aucon obscures he acual mpac of he dvdend paymen. Economerc conclusons vary Maxmzaon of ax (.e., dvdend ax vs. capal gans). Capurng of dvdends However, we can rade dvdends usng vdend waps Forwards Equy waps 6

7 Q U A N T I T A T I V E E E A C H vdends Assumpons on vdends We assume ha he ex-dv daes < < 2 <... are nown and ha we can rade each dvdend n he mare. We also assume ha he dvdends are already adjused of ax and any dscounng o selemen dae,.e. we can loo a he dvdend amoun a he ex-dv dae. The h dvdend s - measureable. Oher Assumpons We have nsananeous deermnsc neres raes r. The equy earns a deermnsc connuous repo-rae m. We assume > ( s sragh forward o ncorporae smple cred rs *,2+ bu we ll sp ha here ) 7

8 Q U A N T I T A T I V E E E A C H vdends oc Prce ynamcs In he absence of frcon cos, he soc prce under rs-neural dynamcs has o fall by he dvdend amoun,.e. In beween dvdend daes, he rs-neural drf under any rsneural measure s gven by raes and repo,.e. we can wre he soc prce beween dvdend daes for : < + as Z ( ) m ds where Z () s a non-negave (local) marngale whch sars n a. : e r s s T : T s he equy growh facor (also called fundng facor ) 8

9 Q U A N T I T A T I V E E E A C H vdends oc Prce ynamcs - Warnng Ths gves Z ( ) Growh rae beween dvdends (Local) marngale par beween dvdends However, specfyng a soc prce dynamc n hs form va Z, does no guaranee ha he soc prce remans posve a he nex dvdend dae: he marngale Z can no have arbrary dynamcs bu needs o be floored o ensure ha he soc prce never falls below any fuure dvdend amoun. 9

10 Q U A N T I T A T I V E E E A C H vdends oc Prce ynamcs - Warnng In oher words, any generc specfcaon of he form dz d r m d ( d) Z does no genercally wor. ee exensve leraure on affne dvdends (specal case cash dvdends) Common fx s o floor he dvdends,.e. ~ : mn{ κ )

11 Q U A N T I T A T I V E E E A C H vdends Towards oc Prce ynamcs wh scree vdends Inuvely, he resrcon s ha he soc prce a any me needs o be above he dscouned value of all fuure dvdends: oherwse, go long soc and forward-sell all dvdends loc-n rs-free reurn. efne herefore he dscouned dvdends as : Ε

12 Q U A N T I T A T I V E E E A C H vdends Theorem (exenson of Buehler 27 [2], Bermudez a al 26 []) The soc prce process remans posve f and only f has he form : ~ X : : Ex-dvdend soc prce where X s a posve local marngale. X s called he pure marngale of he soc prce process. ~ scouned value of fuure dvdends : 2

13 Q U A N T I T A T I V E E E A C H 3 vdends Theorem (exenson of Buehler 27 [2], Bermudez a al 26 []) If X s a rue marngale and f raes are deermnsc, he forward s as usual gven as We can hen re-wre he soc prce process n he form A A X A F : : X F :

14 Q U A N T I T A T I V E E E A C H vdends Proof by consrucon: We wre nsead of n order o ease noaon. Wh hs noaon, we have for : < + Case : n =, we have whch mples ha nce Z s a local marngale wh a posve floor, we can fnd a posve local marngale X () wh un mean such ha Whch n urn gves us ha for ( ) Z () Z () Z : () () Z X E () () X X 4

15 Q U A N T I T A T I V E E E A C H 5 vdends Case 2: =2. for 2 Hence we fnd a bound on he local marngale X () Z () whch mples as before ha we can fnd X () whch sasfes nce s rgh-connuous we conclude ha for 2 es follows by nducon forward. 2 2 () () 2 2 () () 2 2 () Z X Z X Z 2 2 () 2 () Z X 2 2 () () () X Z X () X 2 2 () X

16 Q U A N T I T A T I V E E E A C H vdends Basc rucure of vdend Payng ocs The from : ~ X : has a very clear economc nerpreaon. Implemenaon Exernal approach o modelng dvdends where s a funcon of boh X and, say, some addonal sae u. I does no lend self o dvdends whch are gven as a funcon of soc: posvy of he soc prce can be ensured by mang sure ha he dvdend never exceeds he soc prce. need dfferen represenaon for praccal mplemenaon 6

17 Q U A N T I T A T I V E E E A C H Affne vdends Affne vdends Blac choles Meron: nherenly suppors proporonal dvdends. : ( d Pleny of leraure n more general affne dvdends,.e. : All nown approaches eher Approxmae by approach (.e., he dvdends are no affne). Approxmae by numercal mehods bu hey f well n ou framewor (le us assume raes are deermnsc) ) 7

18 Q U A N T I T A T I V E E E A C H 8 Affne vdends rucure of he oc Prce rec applcaon n our framewor can be done bu s easer o smply wre he proporonal dvdend effecvely as par of he repo-rae m (hs s wha happens n Meron 973 [5]).e. wre All prevous resuls go hrough,.e. we ge : ds r s s e m : ) ( : A A X A F : :

19 Q U A N T I T A T I V E E E A C H Affne vdends Impac on Prcng Vanllas The formula F A X A really means ha we can model a soc prce whch pays affne dvdends by modelng drecly X snce: E K ( F A )EX K T T T whch means ha we can easly compue opon prces on f we now how o compue opon prces on X. T ~ ~ K : K F T A A T T Hence, X can be any classc equy model Blac-choles Heson, AB, l-ab Levy/Affne Numercal Models (LVV) 9

20 Q U A N T I T A T I V E E E A C H Affne vdends Impled Volaly and upre wh Affne vdends The reverse nerpreaon allows us o conver observed mare prces bac no mare prces on X: Call X ~ ( T, K) : whch n urn allows us o Compue X mpled volaly and Compue X mpled local volaly a la upre. mlarly, numercal mehods are very effcen. Varance waps wh Affne vdends PEs Mone-Carlo F T ( F T ee Buehler 27 [2] for a full overvew and ncorporaon of cred. A T ) Call T,( F T A T ~ ) K A T 2

21 Q U A N T I T A T I V E E E A C H 2 vdends ummary Par I A soc prce payng dvdends has he general form In he parcular case of affne dvdends and deermnsc raes, hs can be wren as whch allows very effcen modelng of he equy va X ncludng local volaly. Blac-choles-ype formulas are sragh forward. Anyhng else?... nearly no publshed papers!! X : : ~ ~ : A A X A F : :

22 Par II ochasc Proporonal vdends

23 Q U A N T I T A T I V E E E A C H vdends as an Asse Class Movaon Le us depar from he noon of dvdends on a sngle soc, bu dscuss dvdends of an ndex. A dvdend swap beween T and T 2 smply swaps he realzed dvdends over ha perod (unaccrued) a maury for a fxed sre. : T T 2 K A dvdend fuure s bascally he daly seled correspondng fuure whch seles a he realzed dvdends. oughly f ( T, T ) : T T2 23

24 Q U A N T I T A T I V E E E A C H vdends as an Asse Class 24

25 Q U A N T I T A T I V E E E A C H vdends as an Asse Class Impled vdends From he mare of dvdend swaps, we can mply a fuure level of dvdends. The generally assumed behavor s roughly The shor end s cash (snce raher ceran) The long end s yeld (.e. proporonal dvdends) 25

26 Q U A N T I T A T I V E E E A C H vdends as an Asse Class ochasc Proporonal vdends 26

27 Q U A N T I T A T I V E E E A C H vdends as an Asse Class Producs An (call) opon on dvdends beween T and T 2 pays : T T 2 A dvdend yeld swap approxmaes he yeld from he ndex K : T T2 : monhly / We would prefer, of course T : T T 2 27

28 Q U A N T I T A T I V E E E A C H vdends as an Asse Class Proporonal vdends Before we looed a cash dvdends. However, f he soc prce s srcly posve, we may also wre d ( e ) Ths proporonal dvdend approach maes lfe much easer bascally, o have a nce model, we only have o ensure ha d remans posve. 28

29 Q U A N T I T A T I V E E E A C H vdends as an Asse Class ochasc Proporonal vdends Wha do we wan o acheve: Very effcen model for es-prcng opons on dvdends Blac-choles -ype reference model. Modelng assumpons eermnsc raes (for ease of exposure) We now he expeced dscouned mpled dvdends and herefore he forward F : : Our model should mach he forward and rops a dvdend daes 29

30 Q U A N T I T A T I V E E E A C H vdends as an Asse Class Proporonal vdends Blac & choles d r m d dw ( e dˆ ) ( d ) such ha Fˆ exp dw s s 2 ds 2 s Fˆ exp ( r s m ) dw s s : dˆ o mach he mare forward we chose dˆ log F 3

31 Q U A N T I T A T I V E E E A C H vdends as an Asse Class ochasc Proporonal vdends (Buehler, houb, luys 2 [3]) The frs publshed real sochasc dvdend model: d r m d dw d ( e ) ( d ) wh du d udn db : ( dˆ e u ) c e, dˆ B dvdend facor ochasc offse o he yeld. Correcon erm o f he forward... bu: dvdends can become negave! 3

32 Q U A N T I T A T I V E E E A C H 32 vdends as an Asse Class 2 : E log : T T y

33 Q U A N T I T A T I V E E E A C H vdends as an Asse Class ochasc Proporonal vdends We have Noe 2 F exp sdw s 2 s ds : log /F s normal mean and varance of are analyc. e u c ep : Fnd c such ha E[ ] = F. ep 2: Gven he sochasc dvdend parameers for u, fnd such ha reprces a erm-srucure of mare observable opon prces on. 33

34 Q U A N T I T A T I V E E E A C H vdends as an Asse Class ochasc Proporonal vdends ynamcs The shor-erm dvdend yeld y : log E s approxmaely an affne funcon of u,.e. y a bu A srongly negave correlaon herefore produces very realsc shor-erm behavor (nearly fxed cash ) whle mananng randomness for he longer maures. 34

35 Q U A N T I T A T I V E E E A C H vdends as an Asse Class ochasc Proporonal vdends 35

36 Q U A N T I T A T I V E E E A C H vdends as an Asse Class ochasc Proporonal vdends Good Very fas European opon prcng calbraes o vanllas We can easly compue fuure forwards E [ T ] and herefore also fuure mpled dvdends. Very effcen Mone-Carlo scheme No so grea vdends can become negave Now sew for equy or opons on dvdends. Very lle sew n he opon prces 36

37 Q U A N T I T A T I V E E E A C H vdends as an Asse Class ochasc Proporonal vdends Exensons whn he normal framewor whch wor Jumps on equy ochasc raes Cash dvdend par Whch do no wor AFAWK Local volaly Heson Wha abou non-normal drfs? No soluon nown o us (ye?). Yeld Gap eners negave erory elaed wor: Gaspar s *4+ on erm-srucure models on forwards (HJM-approach). 37

38 Q U A N T I T A T I V E E E A C H vdends as an Asse Class ummary Par II The ochasc Proporonal vdend Model s a frs sep no modelng dvdends beyond he sandard affne framewor. Good model for Blac-choles -ype analyss Two man drawbacs for praccal use: No sew (he bgger ssue) Negave dvdends Wha would be deal Fas model wh full equy sew Ably o modfy vol of dvdends / calbrae qucly vdends behave cash-le when close o ex-dv dae los of wor o do! 38

39 Than you very much for your aenon hp:// *+ Bermudez e al, Equy Hybrd ervaves, Wley 26 *2+ Buehler, Volaly and vdends, WP 27, hp://ssrn.com/absrac=4877 [3] Buehler, houb, luys, ochasc Proporonal vdends, WP 2, hp://ssrn.com/absrac=76758 *4+ Gasper, Fne mensonal Marovan ealzaons for Forward Prce Term rucure Models", ochasc Fnance, 26, Par II, [5] Meron "Theory of aonal Opon Prcng," Bell Journal of Economcs and Managemen cence, 4 (973), pp